Week 3 Problems

These problems are drawn from Chapter 4 of the text, pages 55-58.

  1. (Problem 10) Find all the elements of finite order in $\mathbb{Z}^{\star}$, $\mathbb{Q}^{\star}$, and $\mathbb{R}^{\star}$. What about $\mathbb{C}^{\star}$? (In each case the $\star$ means consider the nonzero elements with multiplication as the group operation.)

  2. (Problem 23) Let $G$ be a group and $a,b\in G$. Prove that:
    • the order of $a$ is the same as the order of $a^{-1}$
    • For any $g\in G$, the order of $gag^{-1}$ is the same as the order of $a$.
    • The order of $ab$ is the same as the order of $ba$.

  3. (Problem 14) Suppose that $G$ is abelian. Prove that, if $a$ and $b$ have finite order in $G$, then so does $ab$. Show that the abelian hypothesis is necessary by considering \(A=\left(\begin{matrix} 0 & 1 \\ -1 & 0\end{matrix}\right)\) and \(B=\left(\begin{matrix} 0 & -1 \\ 1 & -1\end{matrix}\right).\) Show that both $A$ and $B$ have finite order in $\mathrm{GL}_{2}(\mathbb{R})$, but $AB$ does not.

  4. (Problem 26) Prove that $\mathbb{Z}_{p}$ has no nontrivial subgroups if $p$ is prime.

  5. (Problem 30) Suppose that $G$ is a group and that $a,b\in G$. Prove that if $|a|=m$ and $|b|=n$ with $\mathrm{gcd}(m,n)=1$, then \(\langle a\rangle\cap\langle b\rangle=\{e\}.\)

  6. (Problem 38) Prove that the order of an element in a cyclic group $G$ must be a divisor of the order of $G$.

  7. (Problem 24) If $p$ and $q$ are distinct prime numbers, how many generators does $\mathbb{Z}_{pq}$ have?

  8. (Problem 31) Let $G$ be an abelian group. Prove that the subset of elements of $G$ of finite order form a subgroup (this is called the torsion subgroup of $G$). What if $G$ is not abelian? Do the elements of finite order form a subgroup?